An axiomatic construction of an almost full embedding of the category of graphs into the category of R-objects

We construct embeddings G of the category of graphs into categories of R-modules over a commutative ring R which are almost full in the sense that the maps induced by the functoriality of GR[HomGraphs(X,Y)]⟶HomR(GX,GY) are isomorphisms. The symbol R[S] above denotes the free R-module with the basis S. This implies that, for any cotorsion-free ring R, the categories of R-modules are not less complicated than the category of graphs. A similar embedding of graphs into the category of vector spaces with four distinguished subspaces (over any field, e.g. F2={0,1}) is obtained.